Uniqueness of limit cycle in the predator–prey system with symmetric prey isocline

2000 ◽  
Vol 164 (2) ◽  
pp. 203-215 ◽  
Author(s):  
Karel Hası́k
Keyword(s):  
Limit Cycle ◽  
Predator Prey ◽  
Prey System

MULTIPLE BIFURCATIONS IN A PREDATOR–PREY SYSTEM OF HOLLING AND LESLIE TYPE WITH CONSTANT-YIELD PREY HARVESTING

2013 ◽  
Vol 23 (10) ◽  
pp. 1350164 ◽  
Author(s):  
JICAI HUANG ◽  
YIJUN GONG ◽  
JING CHEN
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Phase Portraits ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Prey System ◽  
Multiplicity One ◽  
Multiple Focus ◽  
Parameter Values ◽  
Constant Yield

The bifurcation analysis of a predator–prey system of Holling and Leslie type with constant-yield prey harvesting is carried out in this paper. It is shown that the model has a Bogdanov–Takens singularity (cusp case) of codimension at least 4 for some parameter values. Various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, repelling and attracting Bogdanov–Takens bifurcations of codimensions 2 and 3, are also shown in the model as parameters vary. Hence, there are different parameter values for which the model has a limit cycle, a homoclinic loop, two limit cycles, or a limit cycle coexisting with a homoclinic loop. These results present far richer dynamics compared to the model with no harvesting. Numerical simulations, including the repelling and attracting Bogdanov–Takens bifurcation diagrams and corresponding phase portraits, and the existence of two limit cycles or an unstable limit cycle enclosing a stable multiple focus with multiplicity one, are also given to support the theoretical analysis.

scholarly journals A Resolution of the Paradox of Enrichment

2015 ◽  
Vol 25 (06) ◽  
pp. 1550094
Author(s):  
Z. C. Feng ◽  
Y. Charles Li
Keyword(s):  
Phase Space ◽  
Limit Cycle ◽  
Carrying Capacity ◽  
Dimensionless Number ◽  
Stable Limit ◽  
Predator Prey ◽  
Paradox Of Enrichment ◽  
Infinite Dimensional ◽  
Prey System ◽  
Two Parameters

The paradox of enrichment was observed by Rosenzweig [1971] in a class of predator–prey models. Two of the parameters in the models are crucial for the paradox. These two parameters are the prey's carrying capacity and prey's half-saturation for predation. Intuitively, increasing the carrying capacity due to enrichment of the prey's environment should lead to a more stable predator–prey system. Analytically, it turns out that increasing the carrying capacity always leads to an unstable predator–prey system that is susceptible to extinction from environmental random perturbations. This is the so-called paradox of enrichment. Our resolution here rests upon a closer investigation on a dimensionless number H formed from the carrying capacity and the prey's half-saturation. By recasting the models into dimensionless forms, the models are in fact governed by a few dimensionless numbers including H. The effects of the two parameters: carrying capacity and half-saturation are incorporated into the number H. In fact, increasing the carrying capacity is equivalent (i.e. has the same effect on H) to decreasing the half-saturation which implies more aggressive predation. Since there is no paradox between more aggressive predation and instability of the predator–prey system, the paradox of enrichment is resolved. The so-called instability of the predator–prey system is characterized by the existence of a stable limit cycle in the phase plane, which gets closer and closer to the predator axis and prey axis. Due to random environmental perturbations, this can lead to extinction. We also further explore spatially dependent models for which the phase space is infinite-dimensional. The spatially independent limit cycle which is generated by a Hopf bifurcation from an unstable steady state, is linearly stable in the infinite-dimensional phase space. Numerical simulations indicate that the basin of attraction of the limit cycle is riddled. This shows that spatial perturbations can sometimes (neither always nor never) remove the paradox of enrichment near the limit cycle!

On the Uniqueness of a Limit Cycle for a Predator-Prey System

1988 ◽  
Vol 19 (4) ◽  
pp. 867-878 ◽  
Author(s):  
Lii-Perng Liou ◽  
Kuo-Shung Cheng
Keyword(s):  
Limit Cycle ◽  
Predator Prey ◽  
Prey System

scholarly journals Qualitative Analysis for a Predator Prey System with Holling Type III Functional Response and Prey Refuge

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Xia Liu ◽  
Yepeng Xing
Keyword(s):  
Qualitative Analysis ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Limit Cycle ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Holling Iii Functional Response

A predator prey system with Holling III functional response and constant prey refuge is considered. By using the Dulac criterion, we discuss the global stability of the positive equilibrium of the system. By transforming the system to a Liénard system, the conditions for the existence of exactly one limit cycle for the system are given. Some numerical simulations are presented.

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